A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of linear time-invariant (LTI) systems. Real structures have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not “over-driven”) have behaviour that is close enough to linear so that LTI system theory is an acceptable representation of the input/output behaviour.
Frequency response measurements can be used directly to quantify system performance and substance properties. However time variance of real systems should be considered and therefore the frequency response measurement speed chosen accordingly.
Traditionally sine wave excitation is used for the measurement of frequency domain characteristics (frequency response function—FRF) of different objects and substances.
Simultaneous multisine excitation (composed as a sum of several sine wave components) has been introduced for high speed parallel measurements. Because of a high-value crest factor CF (up to 10 and more without special optimization) the multisine excitation causes serious problems. As the peak value is limited in practical applications (e.g., ±1 V or ±1 mA), the root mean square (RMS) level can be very modest and the excitation energy tends to be too low at each individual excitation frequency (spectral line in the excitation spectrum). Several methods for reducing the crest factor have been presented, the simplest of which—randomizing of initial phases of sine wave components—enables to reduce CF to 1.8. A better idea is to find optimal values for the initial phases of separate sine wave components in the multisine excitation. A better optimization method was developed by A. van den Bos (Schroeder, 1.414×1.17=1.65, van den Bos, 1.414×1.07=1.51, see Tabel 1 in p. 122). Those values are only slightly worse than of a single sine wave (CF=1.414), but still far from ideal (for example, a rectangular signal has CF=1). Van der Ouderaa et al. showed that even CF<1.414 is achievable using iterative calculations—they obtained CF=1.405, for a signal with 15 and 31 equal magnitude components. There is no mathematical expression or ideal algorithm for synthesis of the multisine signal with minimal CF (if all the signal components have equal magnitude). The minimal theoretical value of CF is not known, but CF about 1.5 is achievable in practice. To conclude, a carefully designed multisine signal with optimized CF is the best excitation for fast broadband measurements.
In practice, however, rectangular wave excitations have been introduced as they can be generate by the aid of conventional digital components like triggers, logic gates, and shift registers or computing devices like microcontrollers and signal processors. Moreover, the rectangular waveform can provide more excitation energy compared to sine wave at the same limitations to the peak value (amplitude). For example, the power of a single sine wave is P=A2/2 and its root-mean-square value RMS=A/21/2, where A is the sine wave amplitude. At the same peak value A, the power of a simple rectangular signal is P=A2, and its RMS=A. unfortunately, rectangular signals contain higher harmonics, which complicate measurement procedures and cause serious measurement uncertainties. Several solutions to suppress the role of higher harmonics of rectangular excitations are proposed for the replacement of single sine wave excitations.
To cover a wide frequency band, pseudorandom rectangular waveforms as maximum length sequences (MLS), and rectangular chirp signals are in use. Such signals have one serious disadvantage—their energy is distributed almost equally over the whole frequency band of interest. Therefore, the power spectral density A2/Hz is comparatively low at all the frequencies within the measurement bandwidth. In practice, however, only in some special cases there is a need to measure at all frequencies within the measurement bandwidth, often measuring at 2-3 or up to 100 distinct frequencies is satisfactory. Therefore, it is reasonable to concentrate the energy of excitation signals to certain specific frequencies instead of using uniform energy distribution over all measurement bandwidth. Such the signals are known as multifrequency binary signals. There is mentioned also that such the signals have a severe drawback—the energy is not solely located at the specific frequencies, but is spread also over a great number of undesired, “parasitic”, frequencies.
U.S. Pat. No. 4,093,988 describes a measurement method, in which a pseudo-random maximum length sequence (MLS) of rectangular pulses with equal peak values is used for excitation of the samples under test (electrical or mechanical system). A binary shift register with feedback generates the MLS signal. The response signal is processed with Fast Fourier Transform program running in a computing device. Besides the above described drawback that energy distributes continuously over the full measurement bandwidth, the MLS excitation has two additional disadvantages: 1) the useful excitation bandwidth extends only up to 0.45ƒ of the whole frequency band ƒ with degradation to 50% in power spectral density at 0.45ƒ, and, 2) significant amount of generated energy (near to 40%) falls onto higher frequencies outside the measurement bandwidth 0.45ƒ.
Despite of above described disadvantages, MLS excitation has found an intensive use in audio engineering and electro-acoustic. The MLS excitation is used even more widely, e.g., in impedance spectroscopy.
WO2007/054700 proposes a new type of MLS excitation, where certain higher amplitude rectangular pulses are formed to emphasize the excitation energy within some specific frequency sub-range. A special unit for compensation of transfer function is introduced for that purpose. The proposed solution indeed enables to overcome the main disadvantage of MLS excitation—distribution of excitation energy continuously over all the frequencies of measurement bandwidth—and to enhance the dynamic range of the measurement device in this way. But this invention abandons the ultimate advantage of rectangular waveforms—their minimal crest factor (CF=1). The crest factor CF=A/RMS of the MLS signals by this solution exceeds significantly the unity value, which is the unique property of the sequence of rectangular pulses with equal amplitude.
An alternative to MLS signal is application of rectangular chirp excitation. Rectangular chirp has better energy parameters than MLS, because there is no degradation of the power spectral density within the measurement bandwidth. Moreover, about 90% of the generated energy becomes useful, only 10% of generated energy falls outside the measurement bandwidth and turns useless. But the main disadvantage remains—the energy of excitation is distributed equally over the full measurement bandwidth.
A method for maximizing the overall dynamic range via synthesizing the excitation waveform on the bases of pre-estimation of the frequency response function is described in U.S. Pat. No. 7,194,317. The algorithm of synthesis is based on iterative direct and inverse Fourier transforms of the randomized excitation signal and estimated frequency response function together with their mutual comparison and correction of the excitation signal.
The algorithm is expected to generate the pseudorandom excitation with continuous spectral density. By its essence, the algorithm described in U.S. Pat. No. 7,194,317 is very similar to that method.
A number of attempts are made to adjust the frequency and amplitude properties according to the characteristics of the SUT in order to maximize the overall dynamic range of the calculations by balancing the requirements for the dynamic ranges between the input and output of the SUT.
In one approach the magnitude spectrum is shaped, but the level of the resulting MLS signal is not kept within two discrete levels (+1 and −1).
In another approach the multi-frequency mixed signal in synthesized based on simple superposition of Walsh functions. Resulting binary signal conceals seven primary harmonics (1, 2, 4, 8, 16, 32, and 64) but their amplitudes vary within +/−10%, and are not controllable.
U.S. Pat. No. 7,194,317 describes identification of the systems, where it is desirable to find signals and techniques, which minimize the time spent in data collection. When identifying a multi-input multi-output system, it is also desirable to obtain several statistically uncorrelated signals, thereby making easier to separate out the various input/output relationships of the system, which are measured simultaneously.
The solution suggested here is to delay the original binary multifrequency signal by time, which is long enough that system responses have substantially settled within one effective settling time scale before the cross correlation between any of the signals becomes significant. Therefore these signals are called as nearly stochastically uncorrelated signals.
As the time scale used is one effective settling time, solutions for creating uncorrelated signals are not applicable for experiments, where continuous observation or measurement of the SUT is required.
In general, a limited number of frequencies are needed in order to characterize the SUT in the frequency domain. For example, in majority of practical cases about 2 to 10 frequencies per decade over the whole frequency range of interest describe accurately enough the spectrum of the relatively flat (few dispersions and non resonant behaviour) spectrum. This means, that the spectrally dense MLS and other “white-noise-like” signals are not effective signals to characterize the SUT in frequency domain (transfer function), because as the total energy of the excitation signal is distributed equally over the frequency range, the magnitude of each individual frequency component is low and therefore the initial SNR of the measurements is already low. Therefore it is reasonable to concentrate the available energy within the excitation signal into few limited spectral lines.
For example, the biological samples have quite flat spectrum over the wide frequency range; hence a relatively wide frequency range should be covered by the single multi-frequency excitation signal. This places some strict requirements on the signals, because remarkably different frequencies should be combined into one multi-frequency signal. This results with the signal, consisting of large number of samples, difficult to generate, store and, in particular, to analyze (the calculation time of the DFT increases). Conventionally this is solved by using Short Time Fourier Transform, STFT technique.
The above described methods and algorithms are not applicable for the synthesis of the binary sequence containing predetermined distinct spectral lines which amplitudes are easily controllable (amplitude spectrum shaping) optimizing the overall signal-to-noise ratio (SNR) of the experiments. Additionally, the difficulties to form freely the required frequency content complicate to build test setups for simultaneous, multi-path experiments (orthogonality requirement).
This has great importance in technical fields, where strong limitations to the level of the excitation signals are established in order to prevent extensive influence to the parameters and behaviour of the samples under test (SUT).